Problem 48 - Elementary Algebra

Question

Solve. $$ 33-x=16 $$

Step-by-Step Solution

Answer

The solution is \(x = 17\).

Step 1: Understand the Equation

We are given the equation \(33 - x = 16\). This is a simple linear equation with one variable, \(x\). Our objective is to solve for \(x\).

Step 2: Isolate the Variable

To isolate \(x\) on one side of the equation, we need to remove the constant term (33). We can do this by subtracting 33 from both sides of the equation.

Step 3: Perform the Subtraction

Subtract 33 from both sides of the equation: \(33 - x - 33 = 16 - 33\). This simplifies to \(-x = -17\).

Step 4: Solve for the Variable

To solve for \(x\), we need to get rid of the negative sign. We do this by multiplying both sides by -1: \(-1 \times (-x) = -1 \times (-17)\). This gives us \(x = 17\).

Step 5: Verify the Solution

Substitute \(x = 17\) back into the original equation to verify the solution: \(33 - 17 = 16\). Since both sides equal, our solution is correct.

Key Concepts

Single Variable Equations

Single variable equations are mathematical statements where we solve for one unknown quantity, often represented as \(x\). These equations express a relationship between known numbers and the unknown. For example, in the equation \(33 - x = 16\), \(x\) is the single variable we need to solve for.
The goal in these equations is simple: rearrange the equation to get the variable by itself on one side of the equation. This often involves arithmetic operations to isolate the variable.

Algebraic Manipulation

Algebraic manipulation refers to the process of rearranging and simplifying equations to facilitate solving them. This involves using various arithmetic operations like addition, subtraction, multiplication, and division.
For instance, in the problem \(33 - x = 16\), we subtract 33 from both sides to get \(-x = -17\). This step of subtracting the same number from both sides is crucial for keeping the equation balanced.
After simplifying to \(-x = -17\), the next step is to eliminate the negative sign in front of \(x\). By multiplying both sides of the equation by \(-1\), we obtain \(x = 17\). This process demonstrates how algebraic manipulation can be used to simplify equations and make them easier to solve.

Verifying Solutions

Verifying the solution of an equation is a critical step to ensure the answer is correct. This involves substituting the found variable back into the original equation to see if both sides are equivalent.
For the equation \(33 - x = 16\), we substitute \(x = 17\) to verify: \(33 - 17 = 16\). Since the left side (33 minus 17) equals the right side (16), this confirms our solution is accurate.
This step confirms the logic and arithmetic used in solving the equation are correct, providing confidence in the solution and ensuring no mistakes were made during the algebraic manipulation. It's always a good practice to verify solutions as a final check when solving equations.